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  • 1.
    Amouzgar, Kaveh
    et al.
    Department of Mechanical Engineering, School of Engineering, Jönköping University, Jönköping, Sweden.
    Rashid, Asim
    Department of Mechanical Engineering, School of Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, School of Engineering, Jönköping University, Jönköping, Sweden.
    Multi-objective optimization of a disc brake system by using SPEA2 and RBFN2013In: ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 3B: 39th Design Automation Conference, New York: ASME Press, 2013Conference paper (Other academic)
    Abstract [en]

    Many engineering design optimization problems involve multiple conflicting objectives, which today often are obtained by computational expensive finite element simulations. Evolutionary multi-objective optimization (EMO) methods based on surrogate modeling is one approach of solving this class of problems. In this paper, multi-objective optimization of a disc brake system to a heavy truck by using EMO and radial basis function networks (RBFN) is presented. Three conflicting objectives are considered. These are: 1) minimizing the maximum temperature of the disc brake, 2) maximizing the brake energy of the system and 3) minimizing the mass of the back plate of the brake pad. An iterative Latin hypercube sampling method is used to construct the design of experiments (DoE) for the design variables. Next, thermo-mechanical finite element analysis of the disc brake, including frictional heating between the pad and the disc, is performed in order to determine the values of the first two objectives for the DoE. Surrogate models for the maximum temperature and the brake energy are created using RBFN with polynomial biases. Different radial basis functions are compared using statistical errors and cross validation errors (PRESS) to evaluate the accuracy of the surrogate models and to select the most accurate radial basis function. The multi-objective optimization problem is then solved by employing EMO using the strength Pareto evolutionary algorithm (SPEA2). Finally, the Pareto fronts generated by the proposed methodology are presented and discussed.

  • 2.
    Amouzgar, Kaveh
    et al.
    Department of Mechanical Engineering, School of Engineering, Jönköping University, Jönköping, Sweden .
    Strömberg, Niclas
    Department of Engineering Science, University West, Trollhättan, Sweden.
    An approach towards generating surrogate models by using RBFN with a priori bias2014In: Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, 2014, Vol. 2B, New York, USA: ASME Press, 2014, article id V02BT03A024Conference paper (Refereed)
    Abstract [en]

    In this paper, an approach to generate surrogate modelsconstructed by radial basis function networks (RBFN) with a prioribias is presented. RBFN as a weighted combination of radialbasis functions only, might become singular and no interpolationis found. The standard approach to avoid this is to add a polynomialbias, where the bias is defined by imposing orthogonalityconditions between the weights of the radial basis functionsand the polynomial basis functions. Here, in the proposed a prioriapproach, the regression coefficients of the polynomial biasare simply calculated by using the normal equation without anyneed of the extra orthogonality prerequisite. In addition to thesimplicity of this approach, the method has also proven to predictthe actual functions more accurately compared to the RBFNwith a posteriori bias. Several test functions, including Rosenbrock,Branin-Hoo, Goldstein-Price functions and two mathematicalfunctions (one large scale), are used to evaluate the performanceof the proposed method by conducting a comparisonstudy and error analysis between the RBFN with a priori and aposteriori known biases. Furthermore, the aforementioned approachesare applied to an engineering design problem, that ismodeling of the material properties of a three phase sphericalgraphite iron (SGI) . The corresponding surrogate models arepresented and compared

  • 3.
    Amouzgar, Kaveh
    et al.
    Product Development Department, School of Engineering, Jönköping University, Jönköping, Sweden; School of Engineering Science, University of Skövde, Skövde, Sweden.
    Strömberg, Niclas
    Örebro University, School of Science and Technology. Department of Mechanical Engineering.
    Radial Basis Functions as Surrogate Models with A Priori Bias in Comparison with a Posteriori Bias2017In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 55, no 4, p. 1453-1469Article in journal (Refereed)
    Abstract [en]

    In order to obtain a robust performance, the established approach when using radial basis function networks (RBF) as metamodels is to add a posteriori bias which is defined by extra orthogonality constraints. We mean that this is not needed, instead the bias can simply be set a priori by using the normal equation, i.e. the bias becomes the corresponding regression model. In this paper we demonstrate that the performance of our suggested approach with a priori bias is in general as good as, or even for many test examples better than, the performance of RBF with a posteriori bias. Using our approach, it is clear that the global response is modelled with the bias and that the details are captured with radial basis functions. The accuracy of the two approaches are investigated by using multiple test functions with different degrees of dimensionality. Furthermore, several modeling criteria, such as the type of radial basis functions used in the RBFs, dimension of the test functions, sampling techniques and size of samples, are considered to study their affect on the performance of the approaches. The power of RBF with a priori bias for surrogate based design optimization is also demonstrated by solving an established engineering benchmark of a welded beam and another benchmark for different sampling sets generated by successive screening, random, Latin hypercube and Hammersley sampling, respectively. The results obtained by evaluation of the performance metrics, the modeling criteria and the presented optimal solutions, demonstrate promising potentials of our RBF with a priori bias, in addition to the simplicity and straight-forward use of the approach.

  • 4.
    Christensen, P. W.
    et al.
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Klarbring, Anders
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Pang, Jong Shi
    Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD, USA.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Formulation and comparison of algorithms for frictional contact problems1998In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 42, no 1, p. 145-173Article in journal (Refereed)
    Abstract [en]

    This paper presents two algorithms for solving the discrete, quasi-static, small-displacement, linear elastic, contact problem with Coulomb friction. The algorithms are adoptions of a Newton method for solving B-differentiable equations and an interior point method for solving smooth, constrained equations. For the application of the former method, the contact problem is formulated as a system of B-differentiable equations involving the projection operator onto sets with simple structure; for the application of the latter method, the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non-negativity of variables treated as constraints. The two algorithms are numerically tested for two-dimensional problems containing up to 100 contact nodes and up to 100 time increments. Results show that at the present stage of development, the Newton method is superior both in robustness and speed. Additional comparison is made with a commercial finite element code.

  • 5.
    Gustafsson, Erik
    et al.
    Väderstad-Verken AB, Väderstad, Sweden.
    Hofwing, Magnus
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Residual stresses in a stress lattice: experiments and finite element simulations2009In: Journal of Materials Processing Technology, ISSN 0924-0136, E-ISSN 1873-4774, Vol. 209, no 9, p. 4320-4328Article in journal (Refereed)
    Abstract [en]

    In this work residual stresses in a stress lattice are studied. The residual stresses are both measured and simulated. The stress lattice is casted of low alloyed grey cast iron. In fact, nine similar lattices are casted and measured. The geometry of the lattice consists of three sections in parallel. The diameter of the two outer sections are thinner than the section in the middle. When the stress lattice cools down, this difference in geometry yields that the outer sections start to solidify and contract before the section in the middle. Finally, an equilibrium state, with tensile stresses in the middle and compressive stresses in the outer sections, is reached. The thermo-mechanical simulation of the experiments is performed by using Abaqus. The thermo-mechanical solidification is assumed to be uncoupled. First a thermal analysis, where the lattice is cooled down to room temperature, is performed. Latent heat is included in the analysis by letting the fraction of solid be a linear function of the temperature in the mushy zone. After the thermal analysis a quasi-static mechanical analysis is performed where the temperature history is considered to be the external force. A rate independent J2-plasticity model with isotropic hardening is considered, where the material data depend on the temperature. Tensile tests are performed at room temperature, 200°C, 400°C, 600°C and 800°C in order to evaluate the Young´s modulus, the yield strength and the hardening accurate. In addition, the thermal expansion coefficient is evaluated for temperatures between room temperature and 1000°C. The state of residual stresses is measured by cutting the mid section or the outer section. The corresponding elastic spring-back reveals the state of residual stresses. The measured stresses are compared to the numerical simulations. The simulations show good agreement with the results from the experiments.

  • 6.
    Gustafsson, Erik
    et al.
    Department of Management and Engineering, Linköping University, Linköping, Sweden.
    Hofwing, Magnus
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Simulation and Measurement of Residual Stresses in a Stress Lattice2007In: Proceedings of the 2nd International Conference on Simulation and Modelling of Metallurgical Processes in Steelmaking, Leoben: ASMET , 2007, p. 285-290Conference paper (Refereed)
  • 7.
    Gustafsson, Erik
    et al.
    SweCast AB, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Shape optimization of castings by using successive response surface methodology2008In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 35, no 1, p. 11-28Article in journal (Refereed)
    Abstract [en]

    In this paper, an optimization routine for a thermomechanical problem is presented. The optimization routine is based on the successive response surface methodology where the panning and zooming technique presented by Stander and Craig has been implemented and improved. The optimization routine has been applied to an optimization problem of a three-dimensional beam that undergoes a solidification process. The material in the beam is assumed to be low-alloyed gray iron. The thermomechanical solidification analysis is uncoupled where, first, a thermal analysis is performed to determine the thermal history. This thermal history is then used to calculate the residual stresses in the beam. The residual stresses are solved by using classical J(2)-plasticity with temperature-dependent material properties. The residual stresses from solidification are then carried on to the structural analysis where a mechanical load is applied. These are all linked together via scripts, and the commercial FE software Abaqus is used as the FE solver. The obtained maximum von Mises stress and mass information for every set of parameters are then exported to Matlab where general quadratic response surfaces are fitted by a least square method. Taken together, these response surfaces define a minimum of weight problem, which is solved by using sequential linear programming. To minimize the number of evaluations needed, the parameters are chosen to be D-optimally selected. The numerical results show that the residual stresses from solidification might influence the optimal shape significantly. The residual stress results have been compared with those obtained from casting simulation softwares, and the results are similar. The optimization has been compared with a commercial optimization software and shows very promising results.

  • 8.
    Gustafsson, Erik
    et al.
    Swedish Foundry Association, Jönköping, Sweden.
    Strömberg, Niclas
    School of Engineering, Jönköping University, Jönköping, Sweden.
    Structural optimization of castings by using ABAQUS and Matlab2005In: Competent Design by Castings: Improvements in a Nordic Project / [ed] Jack Samuelsson; Gary Marquis; Jussi Solin, Espoo, Finland: VTT , 2005, Vol. 237, p. 39-52Conference paper (Refereed)
    Abstract [en]

    In this work a general method for structural optimization of nonlinear structures is implemented using FE-analysis. The method utilizes the response surface methodology with polynomial surfaces and nonlinear programming. In such manner a method that is applicable for a large number of different classes of nonlinear problems is obtained. In this paper, the method is utilized to minimize weight of castings by including residual stresses from solidification. This is performed by first determine the residual stresses by a thermomechanical analysis of a metal structure that is cooled from a temperature above liquidus temperature down to room temperature. The thermomechanical analysis is uncoupled where the temperature distribution within the casting as a function of time is determined first and is later on used for residual stress calculations. These residual stresses are then included when the mechanical load is applied to the structure and the problem of minimum of weight is formulated. The structure shown in this paper is an example of a two dimensional geometry. The shape of the structures will of course affect the residual stress distribution during the optimization. The nonlinear models are then solved using ABAQUS/Standard. A set of solutions are generated by solving the model for a pre-defined set of parameters. In order to minimize the number of simulations and still achieve good surface approximations these parameters are taken to be D-optimal. The sets of solutions and parameters are in turn exported to Matlab where general quadratic response surfaces are fitted by the least square method. By utilizing these surfaces the problem of minimum of weight subjected to constraints on stresses is formulated. Finally, the nonlinear optimization problem is solved by sequential linear programming.

  • 9.
    Gustafsson, Erik
    et al.
    Swedish Foundry Association, Jönköping, Sweden.
    Strömberg, Niclas
    School of Engineering, Jönköping University, Jönköping, Sweden.
    Structural optimization of castings by using Abaqus and Matlab2005In: Proceedings of the Abaqus World Users' Conference, 2005Conference paper (Refereed)
    Abstract [en]

    In this work a general method for structural optimization of nonlinear structures is implemented using ABAQUS and Matlab. The method utilizes the response surface methodology with polynomial surfaces and nonlinear programming. In such manner a method that is applicable for a large number of different classes of nonlinear problems is obtained. For instance plasticity problems, thermomechanical problems and contact problems can be optimized using this strategy. In this paper, the method is utilized to minimize weight of castings by including residual stresses from solidification. This is performed by first determine the residual stresses by a thermomechanical analysis of a metal structure that is cooled down from a temperature above liquidus temperature down to room temperature. These residual stresses are then included when the problem of minimum of weight is formulated. The shape of the structure will of course affect the residual stress distribution during the optimization and the optimal shape will be different from the one obtained when residual stresses are not included in the analysis. The method is implemented by using a Python script and m-files. In such way a parameterized model can easily be treated in ABAQUS and Matlab during the optimization process. The parameterized geometry, loads, boundary conditions and mesh are first generated by the ABAQUS/CAE module. The nonlinear models are then solved using ABAQUS/Standard. A set of solutions are generated by solving the model for a pre-defined set of parameters. In order to minimize the number of simulations and still achieve good surface approximations these parameters are taken to be D-optimal. The sets of solutions and parameters are in turn exported to Matlab where general quadratic response surfaces are fitted by the least square method. By utilizing these surfaces the problem of minimum of weight subjected to constraints on stresses is formulated. Finally, the nonlinear optimization problem is solved by sequential linear programming where the linear part is solved using Matlab.

  • 10. Gustafsson, Erik
    et al.
    Strömberg, Niclas
    Department of Mechanical Engineering, University of Jönköping, Sweden.
    Successive Response Surface Methodology by using Neural Networks2007In: Proceedings of the 7th World Congress on Structural and Multidisciplinary Optimization, Seoul: WCSMO-7 Official Agency , 2007Conference paper (Refereed)
  • 11.
    Hofwing, Magnus
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    D-optimality of non-regular design spaces by using a Bayesian modification and a hybrid method2010In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 42, no 1, p. 73-88Article in journal (Refereed)
    Abstract [en]

    In this work a hybrid method of a genetic algorithm  and sequential linear programming is suggested to obtain a D-optimal design of experiments. Regular as well as non-regular design spaces are considered. A D-optimal design of experiments maximizes the determinant of the information matrix, which appears in the normal equation. It is known that D-optimal design of experiments sometimes include duplicate design points. This is, of course, not preferable since duplicates do not add any new information to the response surface approximation and the computational effort is therefore wasted. In this work a Bayesian modification, where higher order terms are added to the response surface approximation, is used in case of duplicates in the design of experiments. In such manner, the draw-back with duplicates might be eliminated. The D-optimal problem, which is obtained by using the Bayesian modification, is then solved by a hybrid method. A hybrid method of a genetic algorithm that generates a starting point for sequential linear programming is developed. The genetic algorithm performs genetic operators such as cross-over and mutation on a binary version of the design of experiments, while the real valued version is used to evaluate the fitness. Next, by taking the gradient of the objective, a LP-problem is formulated which is solved by an interior point method that is available in Matlab. This is repeated in a sequence until convergence is reached. The hybrid method is tested for four numerical examples. Results from the numerical examples show a very robust convergence to a global optimum. Furthermore, the results show that the problem with duplicates is eliminated by using the Bayesian modification.

  • 12.
    Hofwing, Magnus
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    D-optimality of non-regular design spaces by using a Genetic Algorithm2009In: Proceedings of the 8th World Congress on Structural and Multidisciplinary Optimization, International Society for Structural and Multidisciplinary Optimization , 2009Conference paper (Refereed)
  • 13.
    Hofwing, Magnus
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Robustness of residual stresses in castings and an improved process window2010In: Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference 2009: Volume 5, Issue PART B, New York, N.Y.: ASME Press, 2010, p. 1081-1091Conference paper (Refereed)
    Abstract [en]

    In this work the robustness of residual stresses in finite element simulations with respect to deviations in mechanical parameters in castings is evaluated. Young's modulus, the thermal expansion coefficient and the hardening are the studied parameters. A 2D finite element model of a stress lattice is used. The robustness is evaluated by comparing purely finite element based Monte Carlo simulations and Monte Carlo simulations based on linear and quadratic response surfaces. Young's modulus, the thermal expansion coefficient and the hardening are assumed to be normal distributed with a standard deviation that is 10% of their nominal value at different temperatures. In this work an improved process window is also suggested to show the robustness graphically. By using this window it is concluded that least robustness is obtained for high hardening values in combination to deviations in Young's modulus and the thermal expansion coefficient. It is also concluded that quadratic response surface based Monte Carlo simulations substitute finite element based Monte Carlo simulations satisfactory. Furthermore, the standard deviation of the responses are evaluated analytically by using the Gauss formula, and are compared to results from Monte Carlo simulations. The analytical solutions are accurate as long as the Gauss formula is not utilized close to a stationary point.

  • 14.
    Hofwing, Magnus
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Simulation of residual stresses in stamping dies2008In: Proceedings of the IDDRG 2008 Conference: Best in class stamping, 16-18 June 2008, Olofström, Sweden / [ed] Nader Asnafi, Olofström: Industriellt utvecklingscentrum i Olofström AB , 2008, p. 765-776Conference paper (Refereed)
    Abstract [en]

    In the past stamping dies have in principle been designed by rules of thumb and intuition. As the sheet metals in the vehicle industry have got increased mechanical properties in recent years the demands on the stamping dies have increased. For instance increase in stiffness is desirable in order to better control spring-back. The most simple way to satisfy this new demand would be to make the stamping dies even more heavy in order to be able to handle the new sheet metals. Since there are restrictions of the weight of the stamping dies in the stamping machines and since the overhead cranes usually have reached the limit of what they can handle, this is not a desirable solution. Another approach, in order to increase the stiffness without increasing the weight is to use topology optimization. Recently in a master thesis at Volvo Car Corporation a conceptual design of a stamping die has been done by topology optimization. In that work no consideration is taken to the fact that the stamping die is casted. Casting implies that residual stresses possibly are produced during the solidification and cooling process. The residual stresses might affect the fatigue life and the risk of failure of the stamping die.

    In this work the residual stress state after casting is analyzed for the original stamping die as well as the optimized stamping die from the master thesis discussed above. The analyses are performed using an uncoupled approach, where one thermal analysis is followed by a quasi-static elasto-plastic analysis. The thermal analysis simulates the solidification and cooling during the casting process, while the quasi-static elasto-plastic analysis uses the temperature history, obtained from the thermal analysis, in order to build up residual stresses. The thermal analysis includes the release of latent heat. Furthermore, the material properties included in the heat equation (density, conductivity, specific heat) are given as temperature dependent properties for the mould as well as the casting. In the quasi-static elasto-plastic analysis the plasticity is described by the von Mises yield surface in combination with isotropic hardening and the mechanical properties (thermal expansion coefficient, Young's modulus, yield stress, hardening parameter, Poisson's ratio) are given as temperature dependent properties. The simulations show high levels of residual stresses.

     

  • 15.
    Hofwing, Magnus
    et al.
    Department of Mechanical Engineering, University of Jönköping, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, University of Jönköping, Jönköping, Sweden.
    Tapankov, Martin
    Department of Mechanical Engineering, University of Jönköping, Jönköping, Sweden.
    Optimal polynomial regression models by using a genetic algorithm2011In: Proceedings of the second International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering / [ed] Yiannis Tsompanakis; B.H.V. Topping, Kippen, Scotland: Civil-Comp Proceedings , 2011Conference paper (Refereed)
    Abstract [en]

    In this paper a polynomial regression model where the constituents of are of arbitrary order is proposed. A genetic algorithm is used to find the optimal terms to be included in the so-called optimal polynomial regression model . The objective for the genetic algorithm is to minimize the sum of squared errors of the predicted responses. In practice the genetic algorithm generates an optimal set of exponents of the design variables in a polynomial regression model. Several example problems are presented to show the performance and accuracy of the optimal polynomial regression model. Results show a greatly improved performance for optimal polynomial regression models compared to traditional regression models.

  • 16.
    Ireman, Peter J.
    et al.
    Division of Mechanics, Linköping University, Linköping, Sweden.
    Klarbring, Anders
    Division of Mechanics, Linköping University, Linköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Gradient theory of damage coupled to frictional contact and wear, and its numerical treatment2009In: CMES - Computer Modeling in Engineering & Sciences, ISSN 1526-1492, E-ISSN 1526-1506, Vol. 52, no 2, p. 125-158Article in journal (Refereed)
    Abstract [en]

    In this paper finite element approaches for fretting fatigue are proposed on the basis of a non-local model of continuum damage coupled to friction and wear. The model is formulated in the frame-work of a standard material. In a previous paper this was done in the spirit of Maugin, where an extra entropy flux is introduced in the second law in order to include the gradient of the internal variable in a proper manner. In this paper we follow instead the ideas of Frémond and others, where this extra entropy flux is no longer needed, but instead new non-classical balance laws associated to damage, friction and wear, respectively, are derived from the principle of virtual power. The standard material is then defined as usual by state laws based on free energies and complementary laws based on dissipation potentials. In particular, we pick free energies and dissipation potentials that correspond to a non-local continuum damage model coupled to friction and wear. In addition, the boundary conditions at the contact interface creates a coupling between damage and wear. This is a key feature of our model, which makes it very useful in studies of fretting fatigue. By starting from a variational formulation of the governing equations, two different finite element algorithms are implemented. Both algorithms are based on a Newton method for semi-smooth equations. In the first algorithm the Newton method is applied to the entire system of equations, while in the second algorithm the system of equations is split into two different parts such that an elastic wear problem is solved for fixed damage followed by the solution of the damage evolution problem for the updated displacements and contact forces in an iterative process. The latter algorithm can be viewed as a Gauss-Seidel scheme. The numerical performance of the algorithms is investigated for three twodimensional examples of increasing complexity. Based on the numerical solutions, the behavior of the model is also discussed. For instance, it is shown numerically how the initiation of damage depends on the contact geometry, the coefficient of friction and the evolution of wear.

  • 17.
    Ireman, Peter
    et al.
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Klarbring, Anders
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    A model of damage coupled to wear2003In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 40, no 12, p. 2957-2974Article in journal (Refereed)
    Abstract [en]

    The present paper presents a model of damage coupled to wear. The damage model is based on a continuum model including the gradient of the damage variable. Such a model is non-local in the sense that the evolution of damage is governed by a boundary-value problem instead of a local evolution law. Thereby, the well-known mesh-dependency observed for local damage models is removed. Another feature is that the boundary conditions can be used to introduce couplings between bulk damage and processes at the boundary. In this work such a coupling is suggested between bulk damage and wear at the contact interface. The model is regarded as a first attempt to formulate a continuum damage model for studying crack initiation in fretting fatigue.

    The model is given within a thermodynamic framework, where it is assured that the principles of thermodynamics are satisfied. Furthermore, two variational formulations of the full. initial boundary value problem, serving as starting points for finite element discretization, are presented. Finally, preliminary numerical results for a simple one-dimensional example are presented and discussed. It is qualitatively shown how the evolution of damage may influence the wear behaviour and how damage may be initiated by the wear process.

  • 18.
    Ireman, Peter
    et al.
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Klarbring, Anders
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Algorithms for thermoelastic wear problems2002In: Contact mechanics: proceedings of the 3rd Contact Mechanics International Symposiuum / [ed] J.A.C. Martins and Manuel D.P. Monteiro Marques, Dordrecht: Kluwer Academic , 2002, p. 363-370Conference paper (Refereed)
    Abstract [en]

    This contribution concerns the numerical treatment of discrete thermoelastic wear problems. Two different approaches, both utilizing a non-smooth Newton method as non-linear equation solver, are outlined and compared. Furthermore, a numerical example shows how the predicted wear gap is influenced by the bulk properties of the contacting bodies.

  • 19.
    Ireman, Peter
    et al.
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Klarbring, Anders
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Finite element algorithms for thermoelastic wear problems2002In: European journal of mechanics. A, Solids, ISSN 0997-7538, E-ISSN 1873-7285, Vol. 21, no 3, p. 423-440Article in journal (Refereed)
    Abstract [en]

    In the present paper three algorithms are applied to a finite element model of two thermoelastic bodies in frictional wearing contact. All three algorithms utilize a modification of a Newton method for B-differentiable equations as non-linear equation solver. In the first algorithm the fully-coupled system of thermomechanical equations is solved directly using the modified method, while in the other two algorithms the equation system is decoupled in one mechanical part and another thermal part which are solved using an iterative strategy of Gauss–Seidel type. The two iterative algorithms differ in which order the parts are solved. The numerical performance of the algorithms are investigated for two two-dimensional examples. Based on these numerical results, the behaviour of the model is also discussed. It is found that the iterative approach where the thermal subproblem is solved first is slightly more efficient for both examples. Furthermore, it is shown numerically how the predicted wear gap is influenced by the bulk properties of the contacting bodies, in particular how it is influenced by thermal dilatation.

  • 20.
    Klarbring, Anders
    et al.
    Department of Management and Engineering, The Institute of Technology, Linköping University, Linköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    A note on the min-max formulation of stiffness optimization including non-zero prescribed displacements2012In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 45, no 1, p. 147-149Article in journal (Refereed)
    Abstract [en]

    The present theoretical note shows how a naturalobjective function in stiffness optimization, including bothprescribed forces and non-zero prescribed displacements,is the equilibrium potential energy. It also shows how theresulting problem has a saddle point character that may beutilized when calculating sensitivities.

  • 21.
    Klarbring, Anders
    et al.
    Department of Management and Engineering, The Institute of Technology, Linköping University, Linköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Topology optimization of hyperelastic bodies including non-zero prescribed displacements2013In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 47, no 1, p. 37-48Article in journal (Refereed)
    Abstract [en]

    Stiffness topology optimization is usually based on a state problem of linear elasticity, and there seems to be little discussion on what is the limit for such a small rotation-displacement assumption. We show that even for gross rotations that are in all practical aspects small (<3 deg), topology optimization based on a large deformation theory might generate different design concepts compared to what is obtained when small displacement linear elasticity is used. Furthermore, in large rotations, the choice of stiffness objective (potential energy or compliance), can be crucial for the optimal design concept. The paper considers topology optimization of hyperelastic bodies subjected simultaneously to external forces and prescribed non-zero displacements. In that respect it generalizes a recent contribution of ours to large deformations, but we note that the objectives of potential energy and compliance are no longer equivalent in the non-linear case. We use seven different hyperelastic strain energy functions and find that the numerical performance of the Kirchhoff–St.Venant model is in general significantly worse than the performance of the other six models, which are all modifications of this classical law that are equivalent in the limit of infinitesimal strains, but do not contain the well-known collapse in compression. Numerical results are presented for two different problem settings.

  • 22.
    Lundvall, Olle
    et al.
    Department of Mechanical Engineering, Division of Mechanics, Linköping, Sweden; Scania CV AB, Södertälje, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Division of Mechanics, Linköping, Sweden.
    Klarbring, Anders
    Department of Mechanical Engineering, Division of Mechanics, Linköping, Sweden.
    A flexible multi-body approach for frictional contact in spur gears2004In: Journal of Sound and Vibration, ISSN 0022-460X, E-ISSN 1095-8568, Vol. 278, no 3, p. 479-499Article in journal (Refereed)
    Abstract [en]

    In the present paper, a large rotational approach for dynamic contact problems with friction is proposed. The approach is used for modelling a spur gear pair with shafts and bearings. The model is obtained by superposing small displacement elasticity on rigid-body motions, and postulating tribological laws on the gear flanks. The finite element method is used to model the elastic properties of the gear pair. Shafts and bearings are represented by linear springs. The tribological laws of the contact interface are Signorini's contact law and Coulomb's law of friction. An important feature of the approach is that the difficulties of impacting mass nodes are avoided. The governing equations of the model are numerically treated by use of the augmented Lagrangian approach. In such manner the geometry of the gear flanks are well represented in the numerical simulations. It is possible to study accurately the consequences of different types of profile modifications as well as flank errors. In this work, the dynamic transmission error is studied. For instance, it turns out that the effect from profile modification is less significant for the transmission error when frictional effects are included.

  • 23.
    Rashid, Asim
    et al.
    Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Jönköping University, Jönköping, Sweden.
    An Efficient Sequential Approach for Simulation of Thermal Stresses in Disc Brakes2012In: Nordtrib 2012, 15th Nordic Symposium on Tribology, 2012, 2012Conference paper (Other academic)
    Abstract [en]

    In this paper an efficient approach to simulate thermal stresses due to frictional heating of disc brakes is presented. Inthe approach thermal and stress analysis are performed sequentially. The frictional heat analysis is based on the Eulerianmethod, which requires significantly low computational time as compared to the Lagrangian approach. Completethree-dimensional geometries of a disc and a pad are considered for the numerical simulations. The contact forcesare computed at each time step taking the thermal deformations of the disc into account. The nodal temperaturehistory is recorded at each time step and is used in sequentially coupled stress analysis, where a temperature dependentelasto-plastic material model is used to compute the stresses in a disc brake. The results show that during hard braking,high compressive stresses are generated on the disc surface in circumferential direction which cause plastic yielding. Butwhen the disc cools down, the compressive stresses transform to tensile stresses. Such thermoplastic stress history maycause cracks on disc surface after a few braking cycles. These results are in agreement with experimental observationsavailable in the literature.

  • 24.
    Rashid, Asim
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Sequential simulation of thermal stresses in disc brakes for repeated braking2013In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 227, no 8, p. 919-929Article in journal (Refereed)
    Abstract [en]

    In this paper, an efficient sequential approach for simulating thermal stresses in brake discs for repeated braking is presented. First, a frictional heat analysis is performed by using an Eulerian formulation of the disc. Then, by using the temperature history from the first step of the sequence, a plasticity analysis with temperature dependent material data is performed in order to determine the corresponding thermal stresses. Three-dimensional geometries of a disc and a pad to a heavy truck are considered in the numerical simulations. The contact forces are computed at each time step taking the thermal deformations of the disc and pad into account. In such manner, the frictional heat power distribution will also be updated in each time step, which in turn will influence the development of heat bands. The plasticity model is taken to be the von Mises yield criterion with linear kinematic hardening, where both the hardening and the yield limit are temperature dependent. The results show that during hard braking, high compressive stresses are generated on the disc surface in the circumferential direction which cause yielding. But when the disc cools down, these compressive stresses transform to tensile residual stresses. For repeated hard braking when this kind of stress history is repeated, we also show that stress cycles with high amplitudes are developed which might generate low cycle fatigue cracks after a few braking cycles.

  • 25.
    Rashid, Asim
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Thermomechanical Simulation of Wear and Hot Bands in a Disc Brake by Adopting an Eulerian Approach2013In: Eurobrake 2013, 2013Conference paper (Refereed)
    Abstract [en]

    In this paper frictional heating of a disc brake is simulated while taking wear into account. By performing thermomechanical finite element analysis, it is studied how the wear history will influence the development of hot bands. The frictional heat analysis is based on an Eulerian formulation of the disc, which requires significantly lower computational time as compared to a standard Lagrangian approach. A real disc-pad system to a heavy truck is considered, where complete three-dimensional geometries of the ventilated disc and pad are used in the simulations. A sequential approach is adopted, where the contact forces are computed at each time step taking the wear and thermal deformations of the mating parts into account. After each brake cycle, the wear profile of the pad is updated and used in subsequent analysis. The results show that when wear is considered, different distributions of the temperature on disc are obtained for each new brake cycle. After a few braking cycles two hot bands appear on the disc surface instead of only one. These results are in agreement with experimental observations.

  • 26.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    A method for structural dynamic contact problems with friction and wear2003In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 58, no 15, p. 2371-2385Article in journal (Refereed)
    Abstract [en]

    A method for structural dynamic contact problems with friction and wear is suggested. The method is obtained by including wear in the non-smooth contact dynamics method of Moreau. A comparison of the method to the discrete energy-momentum method of Simo and Tarnow is also outlined briefly. The fully discrete equations are treated using the augmented Lagrangian approach, where a non-smooth Newton method is used as the equation solver. Two two-dimensional examples are solved by the method. It is investigated how solutions of contact, friction and wear are influenced by inertia. It is shown that the quasi-static assumption might be questionable for solving contact problems with friction and wear.

  • 27.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    A Newton method for three-dimensional fretting problems1999In: International Journal of Solids and Structures, ISSN 0020-7683, E-ISSN 1879-2146, Vol. 36, no 14, p. 2075-2090Article in journal (Refereed)
    Abstract [en]

    The present paper concerns the numerical treatment of fretting problems using a finite element analysis. The governing equations resulting from a formal finite element discretization of an elastic body with a potential contact surface are considered in a quasi-static setting. The constitutive equations of the potential contact surface are Signorini's contact conditions, Coulomb's law of friction and Archard's law of wear. Using a backward Euler time discretization and an approach based on projections, the governing equations are written as an augmented Lagrangian formulation which is implemented and solved using a Newton algorithm for three-dimensional fretting problems of didactic nature. Details concerning the implementation are provided.

  • 28.
    Strömberg, Niclas
    Örebro University, School of Science and Technology. Örebro University, Örebro, Sweden.
    A Virtual Test Rig for Disc Brake Systems by Adopting a Thermo-Flexible Multi-Body Approach2016In: the 34th annual SAE 2016 Brake Colloquium & Exhibition, Arizona, USA, September 25-28, SAE International , 2016, article id 2016-01-1932Conference paper (Refereed)
  • 29.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    An augmented Lagrangian method for fretting problems1997In: European journal of mechanics. A, Solids, ISSN 0997-7538, E-ISSN 1873-7285, Vol. 16, no 4, p. 573-593Article in journal (Refereed)
    Abstract [en]

    The present work concerns the numerical treatment of fretting in the interface between a body and a rigid foundation. Starting from a variational formulation of a fretting model given in a framework of continuum thermodynamics, an augmented Lagrangian formulation is derived by introducing finite element discretizations in space and a finite difference discretization in time. The augmented Lagrangian formulation is implemented and solved by a Newton method for the two-dimensional case.

  • 30.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    An Efficient Tradeoff Approach for Topology Optimization with Manufacturing Constraints2010In: Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 1: 36th Design Automation Conference, Parts A and B, ASME Press, 2010, p. 1171-1179Conference paper (Refereed)
    Abstract [en]

    In this paper an efficient approach for generating tradeoff curves when performing topology optimization with manufacturing constraints is presented. By minimizing a new stiffness-volume ratio, or in-fact a new compliance-volume product, the tradeoff curve is generated by changing a new design parameter. The volume appearing in the objective is raised to the power of this new design parameter In such manner different conceptual designs can be generated By adopting a nested approach, the problem is easily solved by a simple numerical scheme. This is a nice feature of the approach which makes the numerical performance most efficient and robust. This feature makes it also easy to include manufacturing constraints by simply updating the move limits such that these constraints are satisfied The design parametrization is done by the SIMP-model and patterns of checker-boards are prevented by adopting Sigmund's filter. The efficiency of the approach is demonstrated by presenting tradeoff curves for both 2D- and 3D-problems.

  • 31.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    An Eulerian approach for simulating frictional heating in disc-pad systems2011In: European journal of mechanics. A, Solids, ISSN 0997-7538, E-ISSN 1873-7285, Vol. 30, no 5, p. 673-683Article in journal (Refereed)
    Abstract [en]

    Thermal stresses as a result from frictional heating must be considered when designing disc brakes, clutches or other rotating machine components with sliding contact conditions. The rotational symmetry of the disc in these kind of applications makes it possible to model these systems using an Eulerian approach instead of a Lagrangian framework. In this paper such an approach is developed and implemented. The disc is formulated in an Eulerian frame where the convective terms are defined by the angular velocity. By utilizing the Eulerian framework, a node-to-node formulation of the contact interface is obtained, producing most accurate frictional heat power solutions. The energy balance of the interface is postulated by introducing an interfacial temperature. Both frictional power and contact conductances are included in this energy balance. The contact problem is solved by a non-smooth Newton method. By adopting the augmented Lagrangian approach, this is done by rewriting Signorini’s contact conditions to an equivalent semi-smooth equation. The heat transfer in the disc is discretized by a Petrov–Galerkin approach, i.e. the numerical difficulties due to the non-symmetric convective matrix appearing in a pure Galerkin discretization is treated by following the streamline-upwind approach. In such manner a stabilization is obtained by adding artificial conduction along the streamlines. For each time step the thermo-elastic contact problem is first solved for the temperature field from the previous time step. Then, the heat transfer problem is solved for the corresponding frictional power. In such manner a temperature history is obtained sequentially via the trapezoidal rule. In particular the parameter is set such that both the Crank–Nicolson and the Galerkin methods are utilized. The method seems very promising. This is demonstrated by solving a two-dimensional benchmark as well as a real disc brake system in three dimensions.

  • 32.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    An implicit method for frictional contact, impact and rolling2005In: European journal of mechanics. A, Solids, ISSN 0997-7538, E-ISSN 1873-7285, Vol. 24, no 6, p. 1016-1029Article in journal (Refereed)
    Abstract [en]

    In this paper an implicit method for frictional contact, impact and rolling is suggested. A nonclassical formulation of a two-dimensional hyperelastic body unilaterally constrained to rigid supports is proposed by following the ideas of Moreau and Jean. A total Lagrangian formulation of the system is given. The elastic properties are defined by coupling the second Piola–Kirchhoff stress to the Green–Lagrange strain via the Kirchhoff–St. Venant law. The equation of motion is written in the spirit of Moreau by using the mean value impulses introduced by Jean. The mean value impulses appear explicitly in the equation of motion. In such manner the treatment of nonconstant kinematic transformation matrices becomes straightforward. The rigid supports are described by smooth functions. By utilizing these functions and the mean value impulses, new contact/impact laws of Signorini and Coulomb type are formulated. The governing equations are solved by a nonsmooth Newton method. This is performed by following the augmented Lagrangian approach and deriving the consistent stiffness matrix as well as the contact stiffness matrices. Three two-dimensional examples are solved by the method: a contact problem, an impact problem and a rolling contact problem.

  • 33.
    Strömberg, Niclas
    Department of Mechanical Engineering School of Engineering, Jönköping University, Jönköping, Sweden.
    Contact, impact and rolling of a nonlinear elastic body on rigid foundations2005In: Multibody Dynamics 2005: ECCOMAS Thematic Conference, 2005Conference paper (Refereed)
    Abstract [en]

    In this paper a method for frictional contact, impact and rolling of a two-dimensional hyperelastic body on rigid foundations is presented. A total Lagrangian formulation of the system is given. The elastic properties of the body are defined by coupling the second Piola-Kirchhoff stress to the Green-Lagrange strain via the Kirchhoff-St.Venant law. The rigid supports are described by smooth functions. By introducing the mean value impulses, these functions are utilized to formulate the contact/impact laws. The support functions appear explicitly in the variational formulation of Signorini, and implicitly in the maximal dissipation principle of Coulomb. An attractive property of this approach is that no search algorithm is needed. Another attractive property is that the normal and tangential directions of the supports are well defined. The above constitutive assumptions together with the law of motion, which is written on velocity form, define the governing equations of the system. The governing equations are solved by a nonsmooth Newton method. This is performed by following the augmented Lagrangian approach and deriving the consistent stiffness matrix as well as the contact stiffness matrices. The method is implemented in TriLab. TriLab is a toolbox for simulating contact problems. TriLab is developed using Matlab and Visual Fortran. The Fortran code is linked to Matlab as mex-files. The code is vectorized and the sparsity is utilized. By using TriLab, the presented method will be demonstrated by solving structural dynamic contact problems, impact problems as well as rolling contact problems

  • 34.
    Strömberg, Niclas
    Örebro University, School of Science and Technology. Linköping University.
    Continuum Thermodynamics of Contact Friction and Wear1995Licentiate thesis, monograph (Other academic)
  • 35.
    Strömberg, Niclas
    School of Engineering, Jönköping University, Jönköping, Sweden.
    Development and Implementation of an Eulerian Approach for Efficient Simulation of Frictional Heating in Sliding Contacts2011In: Computational methods for coupled problems in science and engineering IV: proceedings of the IV International Conference on Computational Methods for Coupled Problems in Science and Engineering / [ed] Papadrakakis, M; Onate, E; Schrefler, B, Barcelona: International Center for Numerical Methods in Engineering (CIMNE), 2011, p. 1136-1146Conference paper (Refereed)
    Abstract [en]

    Thermal stresses as a result from frictional heating must be considered when designing disc brakes. The rotational symmetry of a disc brake makes it possible to model this system using an Eulerian approach instead of a Lagrangian framework. In this paper such an approach is developed. The sliding object is formulated in an Eulerian frame where the convective terms are defined by the sliding velocity. A node-to-node formulation of the contact interface is utilized. The energy balance of the interface is stated by introducing an interfacial temperature. Both frictional power and contact conductance are included in this energy balance. The contact problem is solved by a non-smooth Newton method. By adopting the augmented Lagrangian approach, this is done by rewriting Signorini's contact conditions to a system of semi-smooth equations. The heat transfer in the sliding body is discretized by a Petrov-Galerkin approach, i.e. the numerical difficulties due to the non-symmetric convective matrix appearing in a pure Galerkin discretization is treated by following the streamline-upwind approach. In such manner a stabilization is obtained by adding artificial conduction along the streamlines. For each time step the thermoelastic contact problem is first solved for the temperature field from the previous time step. Then, the heat transfer problem is solved for the corresponding frictional power. In such manner a temperature history is obtained via the trapezoidal rule. In particular the parameter is set such that both the Crank-Nicolson and the Galerkin methods are utilized. The method seems very promising. The method is demonstrated for two-dimensional benchmarks as well as a real disc brake system in three dimension.

  • 36.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Finite element treatment of thermoelastic wear problems1998In: IUTAM Symposium on Unilateral Multibody Contacts: Proceedings of the IUTAM Symposium held in Munich, Germany, August 3-7, 1998 / [ed] G.M.L. Gladwell, F. Pfeiffer, Springer, 1998, p. 265-274Conference paper (Refereed)
  • 37.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Finite element treatment of tribological problems2000In: Proceedings of the First International Symposium on Impact and Friction of Solids, Structures and Machines / [ed] Ardéshir Guran, Singapore: World Scientific, 2000, p. 277-280Conference paper (Refereed)
    Abstract [en]

    The present paper concerns the numerical treatment of contact, friction and wear between a thermoelastic body and a rigid foundation. The governing equations of thermoelasticity, Signorini contact, Coulomb friction and Archard wear are put together to a system of equations which is solved by using a Newton method.

  • 38.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    Finite element treatment of two-dimensional thermoelastic wear problems1999In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 177, no 3-4, p. 441-455Article in journal (Refereed)
    Abstract [en]

    The present paper concerns the numerical treatment of thermoelastic wear problems. The governing equations of thermoelasticity coupled to Signorini contact, Coulomb's friction and Archard's wear are formulated as a system of discrete equations. This equation system is solved, using a Bouligand differentiable Newton method, for five problems of didactic nature.

  • 39.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Frictional Contact/Impact between a Hyperelastic Body and Moving Rigid Obstacles2006In: III European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering: Book of Abstracts / [ed] C. A. Motasoares, J. A. C. Martins, H. C. Rodrigues, Jorge A. C. Ambrósio, C. A. B. Pin, Springer, 2006, p. 339-339Conference paper (Refereed)
    Abstract [en]

    In this paper a method for frictional contact/impact between a hyperelastic body and moving rigid obstacles is suggested and investigated. The work is a further development of the suggested method in [1]. The motion of an obstacle is defined by a prescribed translation vector and a prescribed rotation matrix. The geometry of the obstacles are defined by smooth functions. Each function is formulated in a moving frame which is governed by the translation vector and the rotation matrix. These functions are then included in new formulations of Signorini’s conditions and Coulomb’s law of friction. Instead of using contact forces, the mean value impulses are utilized in these formulations, which also are adopted in the law of motion which is given on velocity form. By following this approach, no search algorithm is needed, the normal and tangential directions are well defined and the treatment of non-constant transformation matrices in the law of motion is straight-forward. A total Lagrangian formulation of the system is given. The elastic properties of the body are defined by coupling the second Piola-Kirchhoff stress to the Green-Lagrange strain via the Kirchhoff-St.Venant law. The governing equations are solved by a nonsmooth Newton method. This is performed by following the augmented Lagrangian approach and deriving the consistent stiffness matrix as well as the contact stiffness matrices. The method is implemented in TriLab. TriLab is a user-friendly finite element toolbox for simulating contact and impact problems. TriLab is developed using Matlab and Visual Fortran. The Fortran code is linked to Matlab as mex-files. The code is vectorized and the sparsity is utilized. By using Trilab, the presented method will be demonstrated by solving two-dimensional problems.

  • 40.
    Strömberg, Niclas
    School of Engineering, Jönköping University, Jönköping, Sweden.
    RBDO with Non-Gaussian Variables by using a LHS- and SORM-based SLP approach and Optimal Regression Models2012In: 3rd International Conference on Engineering Optimization, 2012Conference paper (Other academic)
    Abstract [en]

    A general sequential linear programming (SLP) approach for reliability based design optimization (RBDO) with non-Gaussian random variables is presented. The RBDO problems are formulated by using optimal regression models (ORM) as surrogate models and S-optimal design of experiments (DoE). The S-optimal DoE is obtained by maximizing the average mean of the distances between the nearest neighbors. Finite element simulations are performed for the S-optimal DoE and corresponding ORM are obtained by a genetic algorithm. In such manner not only optimal regression coefficients are generated but also optimal rational base functions. The RBDO problems are solved by introducing intermediate variables defined by the iso-probabilistic transformation at the most probable point. By using these variables in the Taylor expansions, a corresponding deterministic linear programming problem is derived, which is corrected by applying second order reliability methods (SORM) as well as Monte Carlo simulations. For low target values on the reliability crude Monte Carlo simulations are used, but for high targets a Latin hypercube sampling (LHS) approach is utilized. The implementation of the suggested sampling- and SORM-based SLP approach is efficient and robust. This is demonstrated by presenting trade-off curves between the objective function, constraints, variables and the target of reliability.

  • 41.
    Strömberg, Niclas
    Örebro University, School of Science and Technology.
    Reliability Based Design Optimization by Using a SLP Approach and Radial Basis Function Networks2016In: Proceedings of the ASME Design Engineering Technical Conference and Computers and Information in Engineering Conference, 2016: Vol. 2B, New York, USA: ASME Press, 2016, p. 309-318Conference paper (Refereed)
    Abstract [en]

    In this paper reliability based design optimization by using radial basis function networks (RBFN) as surrogate models is presented. The RBFN are treated as regression models. By taking the center points equal to the sampling points an interpolation is obtained. The bias of the network is taken to be known a priori or posteriori. In the latter case, the well-known orthogonality constraint between the weights of the RBFN and the polynomial basis functions of the bias is adopted. The optimization is performed by using a first order reliability method (FORM)-based sequential linear programming (SLP) approach, where the Taylor expansions are generated in intermediate variables defined by the iso-probabilistic transformation. In addition, the reliability constraints are expanded at the most probable points which are found by using Newton's method. The Newton algorithm is derived by proposing an in-exact Jacobian. In such manner, a FORM -based LP-formulation in the standard normal space of problems with non-Gaussian variables is suggested. The solution from the LP problem is mapped back to the physical space and the suggested procedure continues in a sequence until convergence is reached. This is implemented for five different distributions: normal, lognormal, Gumbel, gamma and Weibull. It is also presented how the FORM-based SLP approach can be corrected by using second order reliability methods (SORM) and Monte Carlo simulations." In particular the SORM approach of Hohenbichler is studied. The outlined methodology is both efficient and robust. This is demonstrated by solving established benchmarks as well as finite element problems.

  • 42.
    Strömberg, Niclas
    Örebro University, School of Science and Technology.
    Reliability-based design optimization using SORM and SQP2017In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 56, no 3, p. 631-645Article in journal (Refereed)
    Abstract [en]

    In this work a second order approach for reliability-based design optimization (RBDO) with mixtures of uncorrelated non-Gaussian variables is derived by applying second order reliability methods (SORM) and sequential quadratic programming (SQP). The derivation is performed by introducing intermediate variables defined by the incremental iso-probabilistic transformation at the most probable point (MPP). By using these variables in the Taylor expansions of the constraints, a corresponding general first order reliability method (FORM) based quadratic programming (QP) problem is formulated and solved in the standard normal space. The MPP is found in the physical space in the metric of Hasofer-Lind by using a Newton algorithm, where the efficiency of the Newton method is obtained by introducing an inexact Jacobian and a line-search of Armijo type. The FORM-based SQP approach is then corrected by applying four SORM approaches: Breitung, Hohenbichler, Tvedt and a recent suggested formula. The proposed SORM-based SQP approach for RBDO is accurate, efficient and robust. This is demonstrated by solving several established benchmarks, with values on the target of reliability that are considerable higher than what is commonly used, for mixtures of five different distributions (normal, lognormal, Gumbel, gamma and Weibull). Established benchmarks are also generalized in order to study problems with large number of variables and several constraints. For instance, it is shown that the proposed approach efficiently solves a problem with 300 variables and 240 constraints within less than 20 CPU minutes on a laptop. Finally, a most well-know deterministic benchmark of a welded beam is treated as a RBDO problem using the proposed SORM-based SQP approach.

  • 43.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Simulation of Rotary Draw Bending using Abaqus and a Neural Network2005In: Proceedings of Nafems Nordic Conference on Component and System Analysis using Numerical Simulation Techniques - FEA, CFD, MBS, 2005Conference paper (Refereed)
  • 44.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    The Influence of Sliding Friction on Optimal Topologies2013In: Recent Advances in Contact Mechanics: Papers Collected at the 5th Contact Mechanics International Symposium (CMIS2009), Springer Berlin/Heidelberg, 2013, Vol. 56, p. 327-336Conference paper (Refereed)
    Abstract [en]

    In this paper the influence of sliding friction on optimal topologies is investigated and some preliminary results are presented. A design domain unilaterally constrained by a spinning support is considered. Most recently, Stromberg and Klarbring have developed methods for performing topology optimization of linear elastic structures with unilateral contact conditions. In this works sliding friction is also included in the contact model. In such manner it is possible to study how the spinning of the support will influence the optimal design. This was not possible before. The support is modeled by Signorini's contact conditions and Coulomb's law of friction. Signorini's contact conditions are regularized by a smooth approximation, which must not be confused with the well-known penalty approach. The state of the system, which is defined by the equilibrium equations and the smooth approximation, is solved by a Newton method. The design parametrization is obtained by using the SIMP-model. The minimization of compliance for a limited value of volume is considered. The optimization problem is solved by a nested approach where the equilibrium equations are linearized and sensitivities are calculated by the adjoint method. The problem is then solved by SLP, where the LP-problem is solved by an interior point method that is available in the package of Mat lab. In order to avoid mesh-dependency and patterns of checker-boards the sensitivities are filtered by Sigmund's filter. The method is implemented by using Mat lab and Visual Fortran, where the Fortran code is linked to Mat lab as mex-files. The implementation is done for a general design domain in 2D by using fully integrated isoparametric elements. The implementation seems to be very efficient and robust.

  • 45.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Topology Optimization of Bodies in Unilateral Contact by Maximizing the Potential Energy2012In: Proceedings the Eleventh International Conferenceon Computational Structures Technology / [ed] B.H.V. Topping, Kippen, Stirlingshire: Civil-Comp Proceedings , 2012, p. Paper 237-Conference paper (Refereed)
    Abstract [en]

    The modelling of the boundary conditions is crucial in topology optimisation. Small changes in these conditions will typically imply new optimal layouts of material. In many situations the design domain is connected to an assembly of components using contact interfaces. In order to generate proper layouts for this type of design domain one must treat these contact interfaces accurately in the topology optimisation procedure.

    The bottle-neck in topology optimisation of non-linear structural problems, such as contact problems, is to solve the state equations and the adjoint equations. By choosing the potential energy as the objective the latter equations are not needed in the sensitivity analysis. In this paper topology optimization of contact problems including non-zero prescribed displacements by maximizing the potential energy is performed. For contact problems with zero initial contact gaps and zero prescribed displacements this is equivalent to minimising the compliance, which is the standard approach in topology optimisation. However, when the compliance is used as objective in topology optimisation of contact problems an extra adjoint equation must be solved. This is not needed in the formulation presented in this paper.

    The efficiency of the approach is demonstrated by studying three different problems, two problems in two-dimensions and one in a three-dimensional setting. The method is implemented by using Matlab and Intel Fortran, where the Fortran code is linked to Matlab as mex-files. The implementation can be downloaded as a toolbox (Topo4abq). The three problems have beensolved using this toolbox on a laptop with an Intel Core i7 2.67 GHz processor and a 64 bit version of Windows. It is shown that the CPU-time is reduced as much as 25% compared to an adjoint approach developed in previous works.

  • 46.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Topology Optimization of Non-linear Elastic Structures by using SLP2010In: Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference 2009: Volume 5: 35th Design Automation Conference, Parts A and B, ASME Press, 2010, p. 1261-1265Conference paper (Refereed)
    Abstract [en]

    In this paper a method for topology optimization of nonlinear elastic structures is suggested. The method is developed by starting from a total Lagrangian formulation of the system. The internal force is defined by coupling the second Piola-Kirchhoff stress to the Green-Lagrange strain via the Kirchhoff-St. Venant law. The state of equilibrium is obtained by first deriving the consistency stiffness matrix and then using Newton's method to solve the non-linear equations. The design parametrization of the internal force is obtained by adopting the SIMP approach. The minimization of compliance for a limited value of volume is considered. The optimization problem is solved by SLP. This is done by using a nested approach where the equilibrium equation is linearized and the sensitivity of the cost Junction is calculated by the adjoint method. In order to avoid mesh-dependency the sensitivities are filtered by Sigmund'sfilter. The final LP-problem is solved by an interior point method that is available in Matlab. The implementation is done for a general design domain in 2D by using fully integrated isoparametric elements. The implementation seems to be very efficient and robust.

  • 47.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Topology optimization of structures with manufacturing and unilateral contact constraints by minimizing an adjustable compliance–volume product2010In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 42, no 3, p. 341-350Article in journal (Refereed)
    Abstract [en]

    In this paper the concept of extended optimality, or hyperoptimality, is adopted. By following this idea, a new compliance-volume product is suggested as objective. The volume appearing in the product is also raised to the power of a new design parameter which can be set to different values. In such manner design concepts with different volume fractions can be generated by using the approach of extended optimality. Both manufacturing constraints and unilateral contact constraints are included in the proposed method. The manufacturing constraints are implemented by adjusting the move limits such that the draw directions are satisfied. Both one draw direction as well as split draw constraints are considered. The contact conditions are modeled by the augmented Lagrangian approach such that the Jacobian in the Newton algorithm as well as in the adjoint equation becomes symmetric. The design parametrization is done by the SIMP model and Sigmund's filter is utilized when the sensitivities are calculated. The proposed method is very robust and efficient. This is demonstrated by solving problems in both 2D and 3D. The numerical results are also compared to solutions obtained by performing compliance optimization with a constraint on the volume fraction.

  • 48.
    Strömberg, Niclas
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Topology Optimization of Two Linear Elastic Bodies in Unilateral Contact2010In: Proceedings of the 2nd International Conference on Engineering Optimization, 2010Conference paper (Refereed)
    Abstract [en]

    The optimal solutions are most sensitive to the boundary conditions when performing topology optimization of components. In many applications the design domain of the components are subjected to unilateral contact conditions. In order to obtain relevant conceptual designs by topology optimization of such systems, the contact conditions should be included explicitly in the optimization. Recently, in a number of works by Strömberg and Klarbring, such a method has been developed for one elastic body unilateral constrained to rigid supports. Here, this approach is extended such that a system of two elastic bodies in unilateral contact is considered. For this systems the compliance is minimized by adopting the SIMP-model. A nested formulation of the problem is solved by SLP, where the sensitivities are obtained by solving an adjoint equation. In this latter equation, the Jacobian from the Newton method used to solve the state problem appears. The state problem is treated by an augmented Lagrangian formulation of the bodies in contact. Thus, the Jacobian is simply the gradient of the corresponding system of equations to this formulation. The method is implemented in the toolbox Topo4abq by using Matlab and Intel Fortran. The method is both efficient and robust. This is demonstrated by solving several 2D-problems. The results are also compared to the solutions obtained when the contact conditions are treated by joining the two bodies to one body. In a near future3D-problems will also be solved by using the presented approach.

  • 49.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    TriLab: a tool for simulating tribological systems2003In: Proceedings of the 9th International Power Transmission and Gearing Conference, New York: American Society of Mechanical Engineers , 2003, p. 1051-1056Conference paper (Refereed)
    Abstract [en]

    In this paper TriLab is presented. TriLab is a user-friendly finite element tool for simulating tribological systems. TriLab is based on a method for structural dynamic contact problems with friction and wear. The method is reviewed in the paper as well as the quasi-static formulation of frictional contact, the energy-momentum conserving approach of Simo and Tarnow, and Moreau's nonsmooth contact dynamics formulation. TriLab is easy to use and can be utilized for solving both small displacement contact, rolling contact as well as impact problems. This is demonstrated by solving a number of examples.

  • 50.
    Strömberg, Niclas
    Department of Mechanical Engineering, Linköping University, Linköping, Sweden.
    TriLab: a toolbox for tribological systems2003In: Proceedings from the Nordic MATLAB Conference 2003, Copenhagen, Denmark, 2003Conference paper (Refereed)
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